They include geodesics
(paths of shortest distance
such as great circles on a sphere), minimal surfaces (soap films)
and non-linear sigma models in the physics of elementary
particles. They also have applications to the theory of liquid
crystals (see below) and robotics (see, for example
[Y-J Dai, M. Shoji, H. Urakawa,
Click here for a list of books and papers on harmonic maps.
Click here for a list of papers on harmonic morphisms, here for an `atlas of harmonic morphisms', and here for slides of a general talk on harmonic morphisms.
## A description of some of my work and its applications.
Click here for a list of my papers.
In [4] there is a classification of the critical points (points where
38 (1989), 333--340; MR 89m:53013, Zbl.0661.53038]
and to finding meron solutions explicitly
[G. Ghika and M. Visinescu, Meron solutions of the
sigma-model and singular harmonic
maps, Z. Phys. C, 11,
(1982) 353--357; MR 83g:81068].
In [5] it is proved that there is no harmonic map from the torus to the
sphere of degree one; for non-existence results for the Dirichlet problem
see [18,19].
This is of interest in the theory of liquid crystals, see
[K.S. Chou and X.P. Zhu, 42 (1993), 1545--1554; MR 95c:58173, Zbl.0794.53026] and
[J.F. Grotowski, Y. Shen and S. Yan, On various
classes of harmonic maps, Arch. Math. (Basel), 64 (1995),
353--358; MR 96d:58032, Zbl.0817.58009].
In [15] following work of Glaser--Stora and Din--Zakrzewski,
the classification of harmonic maps from the 2-sphere to complex
projective space is carefully described; for maps into
certain other symmetric spaces, see [16,22,24, 25,27,28,31]. This is
the non-linear sigma-model of elementary particle physics. For some
applications, see
[G. Dunne, 300 (1988), 223--237;
MR 90f:81064].
In [45] we describe the space of harmonic 2-spheres in S^{2} to CP^{2}, thus giving the nullity of any such
harmonic map; it also has bearing on the behaviour of weakly harmonic E-minimizing maps from a 3-manifold to CP^{2} near a singularity and the structure of the singular set of such maps from any manifold to CP^{2}.
In [26] and [30] we construct all harmonic morphisms from
In [37] for maps from an Einstein 4-manifold to a surface it is shown that
the existence of a harmonic morphism implies the existence of a
Hermitian structure, an inverse construction is given in the anti-self
dual case; see
[V. Apostolov and P. Gauduchon,
In [38] a framework for multivalued harmonic morphisms is given which generalizes the idea of multivalued complex analytic functions to higher dimensional domains.
In [44,48] we construct the first examples of local and global
complex-valued harmonic morphisms
from Euclidean spaces
In [ [ University of Leeds ] [ School of Maths. ] [ Pure Maths ] [ J.C. Wood ] This page is maintained by
J.C. Wood Last Updated 7 November 2003 |